[Retracted] On Fundamental Algebraic Characterizations of <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5268pt" id="M1" height="12.3952pt" version="1.1" viewBox="-0.0657574 -7.8684 9.691 12.3952" width="9.691pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-234" d="M538 96L524 119C492 88 454 63 446 63C439 63 434 70 440 101C463 223 491 341 518 448H508L433 422L401 276C355 192 240 56 188 56C163 56 154 89 163 133C184 233 207 338 235 443L230 448L152 424L58 17C40 -60 23 -143 23 -185C23 -241 42 -261 63 -261C92 -261 117 -241 125 -227L124 -221C109 -209 89 -165 89 -103C89 -55 96 -13 105 26H107C121 -3 134 -12 151 -12C172 -12 194 -5 221 16C279 61 330 124 384 194H386C381 159 377 141 368 105C343 3 363 -12 383 -12C416 -12 486 35 538 96Z"/></g></svg>-</span>Fuzzy Normal Subgroups

Masmali, Ibtisam; Shuaib, Umer; Razaq, Abdul; Fatima, Areeba; Alhamzi, Ghaliah

doi:https://doi.org/10.1155/2022/2703489

Journal of Function Spaces

On this page

Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Authors’ Contributions References Copyright Related Articles

Research Article Retraction

!

This article has been Retracted. To view the article details, please click the ‘Retraction’ tab above.

Special Issue

Fuzzy Sets and Their Applications in Mathematics

View this Special Issue

Research Article | Open Access

Volume 2022 | Article ID 2703489 | https://doi.org/10.1155/2022/2703489

[Retracted] On Fundamental Algebraic Characterizations of -Fuzzy Normal Subgroups

Ibtisam Masmali,¹Umer Shuaib,²Abdul Razaq,³Areeba Fatima,²and Ghaliah Alhamzi⁴

Academic Editor: Mohammed S. Abdo

Received12 Mar 2022

Revised17 Apr 2022

Accepted30 Apr 2022

Published19 May 2022

Abstract

In this article, we present the study of -fuzzy subgroups and prove numerous fundamental algebraic attributes of this newly defined notion. We also define the concept of -fuzzy normal subgroup and investigate many vital algebraic characteristics of these phenomena. In addition, we characterize the quotient group induced by this particular fuzzy normal subgroup and establish a group isomorphism between the quotient groups and . Furthermore, we initiate the study of level subgroup, open level subgroup, and tangible subgroup of a -fuzzy subgroup and emphasize the significance of -fuzzy normal subgroups by establishing a relationship between these newly defined notions and -fuzzy normal subgroup.

1. Introduction

The theory of fuzzy logic is based on the concept of relative graded membership as inspired by the processes of human perception. This logic deals with information that is uncertain, imprecise, vague, partly true, or without clear boundaries. Moreover, this theory provides a mathematical framework within which ambiguous conceptual phenomena can be studied with precision. New computing methods based on fuzzy logic can be used in the development of intelligent system for decision-making, identification, pattern recognition, optimization, and control system. This particular logic is currently being used in the industrial practice of advanced information technology. One of the most important applications of group theory is its key role in geometry and cryptography. Geometry is the study of properties of a space that are invariant under a group of transformations of that space. The theory of groups is used to classify the symmetries of molecules, crystal structures, and regular polyhedral. It is also used to solve the old issues of algebra.

Fuzzy subsets (FSs) have a central position in modern mathematics. In 1965, Zadeh [1] proposed the idea of FSs. The idea of fuzzy subgroup (FSG) was presented by Rosenfeld [2] in the framework of FSs in 1971. Das [3] defined the level subgroups (LSGs) of a FSG in 1981. Mukherjee and Bhattacharya [4] characterized the notions of fuzzy normal subgroup (FNSG) and fuzzy coset in 1984. Mashour et al. [5] described many important properties of FNSGs in 1990. The characterizations of fuzzy conjugate subgroups and fuzzy characteristic subgroups by their level subgroups were presented in [6]. To see more on the development of theory of FSGs, we refer to [7–11]. The fuzzy logic was effectively used in industrial management [12], coding theory [13] and forecasting systems [14]. Rowlands and Wang [15] applied fuzzy logic to present a useful technique to design fuzzy-SPC evaluation and control method in 2000. Liu [16] developed a new method of constructing quotient groups induced by FNSGs and proved the corresponding isomorphism theorem. In 2005, Mordeson et al. [17] discussed the zero power of a FSG along with various important analytical characteristics of FSGs. Li et al. [18] gave the efficient solutions of linear programming problems based on fuzzy logic. Faraz and Shapiro [19] designed a fuzzy statistical control chart that explains the existence of fuzziness in data in 2010. The authors proposed a new type of FNSGs and fuzzy cosets [20]. Celik et al. [21] introduced the concept of fuzzy soft rings and studied some of their structural properties. Borah and Hazarika [22] innovated the study of mixed fuzzy soft topology with separation axioms along with applications of fuzzy soft sets in chemistry. In addition, useful characteristics of FSGs in various algebraic structures can be viewed in [23–27]. Ullah et al. [28] utilized the idea of complex Pythagorean FS in pattern recognition problems. Alghazzawi et al. [29] introduced anti-intuitionistic fuzzy subgroups and presented a comprehensive study of this concept. In [30] Bejines et al. investigated various properties of aggregation of fuzzy subgroups.

We recommend the readers to study [31–39] to obtain more information on this topic.

The fuzzy sets have ability to play an effective role to solve many physical problems. These sets provide us the meaningful representations of measuring uncertainty. Despite all of these advantages, we still face vast complications to counter various physical situations. This motivates us to define the notion of -fuzzy set through which one can have multiple options to investigate a specific real-world situation in much efficient way by choosing appropriate value of the parameter .

The -fuzzy sets are capable to deal with the uncertainty and vagueness of a physical problem much more effectively than with the theory of the classic fuzzy set, especially in the area of logic programming and decision-making, financial services, psychological examinations, medical diagnosis, career determination, and artificial intelligence. For instance, -fuzzy sets are used to improve the resolution of a certain picture. A photograph of a person describes his many biological features like tall or short, heavy or thin, old or young, and male or female. Sometimes this picture becomes corrupted due to many distortions in the lenses. Such distortions are removed under the framework of these particular fuzzy sets by choosing the appropriate value of the parameter .

In this article, we initiate the study of -FSs and their level sets along with many set theoretical properties of these phenomena. We also propose the concept of -FSG based over a -FS and investigate some of their various algebraic properties. We extend this ideology by defining the notions of -FNSG and -fuzzy coset and establish a group isomorphism between the quotient groups of these newly defined ideas. In addition, we characterize the concept of level subgroup (LSG), open level subgroup (OLSG), and tangible subgroup (TSG) of a -FSG and establish a relationship between a -FNSG and the above stated notions.

After a brief discussion about the historical background and significance of FSG, the rest of the article is organized as follows: the second section contains a brief review of some fundamental definitions of basic notions which are quite useful to understand the novelty of this study. Section 3 is dedicated to introduce the notion of -FSG defined over a -FS, and many fundamental algebraic characteristics of this notion are investigated. In Section 4, we extend the study of -FSGs by defining the notions of -FNSG and quotient group induced by this particular FNSG. Furthermore, we establish a group isomorphism between the quotient groups and . In addition, we define the concepts of LSG, OLSG, and TSG of a -FSG and shed light the importance of -FNSG by developing an important relationship between these concepts and -FNSG.

2. Preliminaries

In this section, we study some fundamental concepts of FSGs which are quite essential to obtain the basic group theoretic outcomes in terms of their respective fuzzy counterparts.

Definition 1 (see [1]). Consider an element of a universe . Any function from to the closed unit interval is called a fuzzy set.

Remark 2 (see [17]). Any two FSs and of a universe admit the following characteristics. For any (1) iff (2) iff and (3)The complement of the FS is and is determined as (4)(5)

Definition 3 (see [17]). The -cut and strong -cut of a FS are denoted by and , respectively, and are defined in the following ways: (1)(2)

Definition 4 (see [17]). The support of an FS is denoted by and is described as

Definition 5 (see [17]). An FS of a group is called FSG of if it satisfies the following conditions: (i)(ii), for all

Definition 6 (see [4]). The FSG of a group is said to be fuzzy normal subgroup of if for all .

Definition 7 (see [4]). Let be FSG and be a fixed element of . Then, fuzzy left coset of in is denoted by and is interpreted in the following way: , for all . The fuzzy right coset is determined in the same manner.

Definition 8 (see [8]). Let and be two FSs of groups and _, respectively, and be a mapping. Then, and are defined as , and for every

3. Algebraic Properties of -Fuzzy Subgroups

This section is committed to initiate the conception of -FSG defined over -FS. We study several algebraic aspects of these phenomena.

Definition 9. Let be a FS of universe and . Then, FS is called the -fuzzy set of universe w.r.t. and is defined as , for all , where the algebraic sum is described as

Remark 10. (1)Evidently, we obtain the classical FS from -FS for whereas the case becomes crisp set for the value of (2)Let and be any two -FSs of and , respectively, and be a mapping. Then(3)(4)

Definition 11. For any -FS and , we denoted -cut of by such that

Definition 12. Let be a -FS and . The strong -cut of is denoted by and is described as

Example 1. (1)An important application of -FS in the medical decision-making is to determine the amount of the drug dose for patients by setting an appropriate value of the parameter related to age and weight of the patients. The radiation therapy can be improved in the framework of -FSs by selecting an appropriate value of (2)Another important significance of -FSs can be viewed in the formulation of default timings for the traffic lights as it effectively supports the fuzzy controller to adjust the timings depending upon the value of the parameter

Definition 13. (1)The union of -FSs and is denoted by and is defined as follows: (2)The intersection of -FSs and is denoted by and is interpreted as follows: (3)The complement of -FS is denoted by and is describe as follows:

Definition 14. Let be an FS of a group and . Then, is called a -fuzzy subgroup of if satisfies the following conditions: (1)(2), for all The above conditions can also be defined in the following way.

Remark 15. Every -FSG of a group admits the following characteristics (a), for all (b) which implies that (2)Every FSG of is a -FSG of but the converse is not true, in general(3)The intersection of a family of -FSGs of is also a -FSG of (4)The union of any two -FSGs of need not be a -FSG of (5)In the following result, we establish some important relationships between -cut and strong -cut of -

Theorem 16. Let and be any two -FS; then, the following attributes hold for any (i) implies that and (ii)(iii) and (iv) and

Proof. (i)In view of Definition 12, for any element , . It means that . Thus, (ii)Let , then by using Definition 11, we have . This implies that . Consequently, Similarly, the assertion can be proved for strong cut of . (iii)Let , then . It follows that . This means that . Therefore, Hence Now suppose, then . This implies that ; therefore, . It follows that Hence From (5) and (6), we obtain . (iv)Let then This implies that , so and . Therefore, , and consequently, The application of efinition 11 on yields that and . This shows that . Consequently, From (7) and (8), we obtain the required equality.
The second part of (iv) can be proved in the same manner. (v)For any element , we obtain . Therefore, . This means that or . Therefore, , and consequently, The application of Definition 12 on yields that or . This shows that Consequently, By comparing the relations (9) and (10), we get the required result.
The other part of (v) can be proved in the same manner.
The following theorem presents a necessary and sufficient condition for a -FS to be a -FSG.

Theorem 17. A -FS is a -FSG of a group if and only if is a subgroup of for each , where is the identity element of .

Proof. Suppose is a -FSG and , then in the view of Definition 11, we have and . This implies that Consequently, . Moreover, for , we have , implying that for all . This means that , and hence, is a subgroup of .
Conversely, assume that is a subgroup of . For any , let and . Then, clearly, and . Suppose , then . Since and is a subgroup of , therefore, . Then, implying that . For any , let . Then, and This implies that , and hence, is a -FSG of .

4. Fundamental Algebraic Characterizations of -Fuzzy Normal Subgroups

In this section, we categorize the ideas of -fuzzy cosets and -FNSGs and prove fundamental algebraic characteristics of these phenomena. We also define the quotient group induced by -FNSG and obtain a group isomorphism between and . Furthermore, we innovate the study of the concepts of LSG, OLSG, and TSG of a -FSG and develop a relationship of these notions with -FNSG.

4.1. Definition

Let be a -FSG and be any fixed element of a group Then, -fuzzy left coset of in is represented by and is defined as follows:

Likewise, the notion of -fuzzy right coset can be established.

4.2. Definition

A -FSG is called -fuzzy normal subgroup of if for all .

In other words, a -FNSG can also be defined as , for all

Theorem 18. The following statements are equivalent for any -FSG of a group (i) is a -FNSG of (ii)(iii)(iv)for all

Proof. i ii: For any two elements we have .
ii iii: The required inequality is a straightforward implication of ii.
iii iv: Consider .
In other words, .
iv i: Consider Moreover, The required result can be acquired by using (12) and (13).
In the following theorem, we find a necessary and sufficient condition for a -FSG to be -FNSG.

Theorem 19. A -FSG is a -FNSG of a group if and only if is a normal subgroup of for each , where is the identity element of .

Proof. Suppose is a -FNSG. The application of Definition 11 on , we have and . Consider . Consequently, , and hence, is a normal subgroup of .
Conversely, assume that is a normal subgroup of . Let and , then clearly, . Since is a normal subgroup of therefore, . This means that . It follows that satisfies fact iii of Theorem 18 and according to Theorem 20 is a -FNSG of .

4.3. Definition

The support of a -FS is denoted by and is described as .

In the following result, we show that support of is a normal subgroup of .

Theorem 20. Let be a -FNSG of a group . Then, is normal subgroup of .

Proof. The application of Definitions (9) and (15) on , we have . Consequently, ; therefore, is a subgroup of Moreover, by applying the normality of , we have , for any and . Consequently, ; thus, is a normal subgroup of .
In the following result, we show that is normal subgroup of .

Theorem 21. Let be a -FNSG of a group ; then, the set is a normal subgroup of

Proof. The application of Definition (9) on , we have We also know that By the comparison of (15) and (16), we get .
Hence, is a subgroup of .
Moreover, by applying the normality of , we have , for any and . Consequently, ; this shows that is a normal subgroup of

The following example describes the fact that leads to note that the converses of the Theorems 20 and 21 are not true.

Example 2. Consider a subgroup of the dihedral group . Clearly, is not normal in . The FS of is as follows: The -FS of correspond to the value is given by Then, and . Obviously, and are normal subgroups of . But is not a FNSG of because .

The following result describes another approach to obtain a -FNSG.

Theorem 22. A -FSG is a -FNSG of a group if and only if is constant on the conjugate classes of

Proof. Suppose is a -FNSG of . Then This means that is a constant on the conjugate classes of
Conversely, suppose that is constant on each conjugate class of . Then, for all Consequently, is a -FNSG of .

Theorem 23. Every -FNSG of admits the following properties for all : (i)(ii)

Proof. (i) Suppose that , where . By using Definition (12) in the above relation, we obtain So, . This means that , and ultimately, we have Similarly By comparing the relations (21) and (22), we get the required equality.
Conversely, let . Then, and . By applying Definition (12) on , we have which implies that Similarly The required equality can be obtained by using (24) and (25).
(ii) This part proved in the same way.

Theorem 24. Let be a -FNSG of a group and ; then, , for all .

Proof. Let and ; then, . This implies that . Consider This means that Likewise, In view of (27) and (28), we obtain .

Theorem 25. Let be a -FNSG of ; then, for all , and implies

Proof. The application of Theorem 23 and using the given condition on , we have . Consider which implies that . It follows that ; therefore Similarly By comparing the relations (30) and (31), we get the required equality.

Theorem 26. Suppose that is a -FNSG of and is the set of all -fuzzy cosets of in . Define a binary operation on in the following way: , where . Then, forms a group.

Proof. Let and such that and . Then, for each, we have and . Moreover, Since and therefore and . Particularly, we have Thus, .
Also, we know that . This implies that Similarly, we have By comparing (33) and (35), we have Thus, . Hence, is well defined. (1)Furthermore, is associative(2) is inverse of (3) is identity element in Hence, is a group.
Note that is known as the -fuzzy quotient group induced by .
In the following result, we establish a group isomorphism between and .

Theorem 27. Let -FNSG. Then, .

Proof. The application of Theorem 19 and using the fact that FNSG, we have that is a normal subgroup of . This ensures the existence of . Define a map by the rule , . Consider In view of Theorem 23, the above relation becomes This implies that This shows that is a well defined mapping.
The function is injective because implies that By applying heorem 23, we have Clearly, is surjective because for each , there exists such that .
Moreover, is homomorphism as for each , Consequently, is a group isomorphism between and .
In the upcoming theorem, we establish a natural homomorphism between the group and its quotient group by -FNSG .

Theorem 28. Let -FNSG and . Then, the map defined by is a natural homomorphism with its kernel .

Proof. Let , then It means that is a homomorphism.
Also

Definition 29. Let and be any two -FSGs and be a fixed element of a group Then, is -fuzzy conjugate to if there exists an element such that . If such is the case, then we write , where

Theorem 30. For any -FSG of , then, is a -FNSG of and is the largest -FNSG of that is contained in .

Proof. Given that is a -FSG and be any element of , in view of Remark 15, we have that is a -FSG of . Note that, for all , Thus, for all . Hence, by Theorem 18, is a -FNSG of .
Now let be a -FNSG of with . Then, for all . So . Therefore, is the largest -FNSG of that is contained in .

Definition 31. Let be a -FSG of and be the tip of . Then (i)The level subgroup of is denoted by and is defined as follows: (ii)The open level subgroup of is denoted by and is defined as follows: (iii)The tangible subgroup of is denoted by and is defined as follows:

Example 3. Consider the quaternion group of order 8. The FS of is defined as follows: The -FS of correspond to the value is given by Note that the tip of is and (i)The LSG corresponding to the value is given by (ii)The OLSG corresponding to the value is given by (iii)The TSG corresponding to the value is given by Clearly, , , and are the subgroups of

The following result describes another approach to obtain a -FNSG.

Theorem 32. For any -FSG of a group , the following statements are equivalent: (i) is a -FNSG in (ii)Every LSG of is normal in (iii)Every TSG of is normal in (iv)Every OLSG of is normal in

Proof. i ii: The application of Theorem 19 ensures the validity of i ii.
ii iii: Since every TSG of is a LSG of which is normal in . Thus, every TSG of is normal in .
iii iv: Let , where is the tip of . Then, which is normal in , being the union of an increasing chain of normal subgroups of .
iv i: If possible, let there exist two elements such that . Then, , since is normal in . Therefore, . This implies that , which is not true.

5. Conclusions

In this work, we introduced the notion of -FSG defined over -FS and studies various important properties. We also presented the concept of -FNSGs and proved some important related results. Furthermore, we have defined the quotient group and proved the existence of isomorphism between and In addition, we have defined the notions of LSG, OLSG, and TSG of a -FSG and have developed a relationship between these notions and -FNSG. We are hopeful this article provides a base to study many more important group theoretic topics like isomorphism theorems, Cayley’s theorem, Lagrange’s theorem, Sylow’s theorem, and many others in-fuzzy environment.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

All authors have contributed equally to this paper in all aspects. All authors have read and agreed to the published version of the manuscript.

References

L. A. Zadeh, “Fuzzy sets. Information and control,” Inform Control, vol. 8, no. 3, pp. 338–353, 1965.
View at: Publisher Site | Google Scholar
A. Rosenfeld, “Fuzzy groups,” Journal of Mathematical Analysis and Applications, vol. 35, no. 3, pp. 512–517, 1971.
View at: Publisher Site | Google Scholar
P. S. Das, “Fuzzy groups and level subgroups,” Journal of Mathematical Analysis and Applications, vol. 84, no. 1, pp. 264–269, 1981.
View at: Publisher Site | Google Scholar
N. P. Mukherjee and P. Bhattacharya, “Fuzzy normal subgroups and fuzzy cosets,” Information Sciences, vol. 34, no. 3, pp. 225–239, 1984.
View at: Publisher Site | Google Scholar
A. S. Mashour, H. Ghanim, and F. I. Sidky, “Normal fuzzy subgroups,” Information Sciences, vol. 20, pp. 53–59, 1990.
View at: Google Scholar
V. N. Dixit, R. Kumar, and N. Ajmal, “Level subgroups and union of fuzzy subgroups,” Fuzzy Sets and Systems, vol. 37, no. 3, pp. 359–371, 1990.
View at: Publisher Site | Google Scholar
D. S. Malik, J. N. Mordeson, and P. S. Nair, “Fuzzy normal subgroups in fuzzy subgroups,” Journal of the Korean Mathematical Society, vol. 29, no. 1, pp. 1–8, 1992.
View at: Google Scholar
S. Ray, “Isomorphic fuzzy groups,” Fuzzy Sets and Systems, vol. 50, no. 2, pp. 201–207, 1992.
View at: Publisher Site | Google Scholar
N. Ajmal, “The lattice of fuzzy normal subgroups is modular,” Information Sciences, vol. 83, no. 3-4, pp. 199–209, 1995.
View at: Publisher Site | Google Scholar
M. A. Mishref, “Normal fuzzy subgroups and fuzzy normal series of finite groups,” Fuzzy Sets and Systems, vol. 72, no. 3, pp. 379–383, 1995.
View at: Publisher Site | Google Scholar
J. Zhou, S. Y. Li, and S. Y. Li, “LP-fuzzy normal subgroups and fuzzy quotient groups,” Journal of Fuzzy Mathematics, vol. 5, pp. 27–40, 1997.
View at: Google Scholar
A. J. Van der Wal, “Application of fuzzy logic control in industry,” Fuzzy Sets and Systems, vol. 74, no. 1, pp. 33–41, 1995.
View at: Publisher Site | Google Scholar
S. C. Cheng and J. N. Mordeson, “Applications of fuzzy algebra in automata theory and coding theory,” in Proceedings of IEEE 5th International Fuzzy Systems, pp. 125–129, New Orleans, LA, USA, Sept. 1996.
View at: Publisher Site | Google Scholar
M. Y. Chow and H. Tram, “Application of fuzzy logic technology for spatial load forecasting,” in Proceedings of 1996 Transmission and Distribution Conference and Exposition, pp. 608–614, Los Angeles, CA, USA, Sept. 1996.
View at: Publisher Site | Google Scholar
H. Rowlands and L. R. Wang, “An approach of fuzzy logic evaluation and control in SPC,” Quality and Reliability Engineering International, vol. 16, no. 2, pp. 91–98, 2000.
View at: Publisher Site | Google Scholar
Y. L. Liu, “Quotient groups induced by fuzzy subgroups,” Quasigroups and Related Systems, vol. 11, pp. 71–78, 2004.
View at: Google Scholar
J. N. Mordeson, K. R. Bhutani, and A. Rosenfeld, Fuzzy Group Theory, Springer, New York, 2005.
View at: Publisher Site
X. Q. Li, B. Zhang, and H. Li, “Computing efficient solutions to fuzzy multiple objective linear programming problems,” Fuzzy Sets and Systems, vol. 157, no. 10, pp. 1328–1332, 2006.
View at: Publisher Site | Google Scholar
A. Faraz and A. F. Shapiro, “An application of fuzzy random variables to control charts,” Fuzzy Sets and Systems, vol. 161, no. 20, pp. 2684–2694, 2010.
View at: Publisher Site | Google Scholar
S. Abdullah, M. Aslam, T. A. Khan, and M. Naeem, “A new type of fuzzy normal subgroups and fuzzy cosets,” Journal of Intelligent & Fuzzy Systems, vol. 25, no. 1, pp. 37–47, 2013.
View at: Publisher Site | Google Scholar
Y. Celik, C. Ekiz, and S. Yamak, “Applications of fuzzy soft sets in ring theory,” Annals Fuzzy Mathematics and Informatics, vol. 5, no. 3, pp. 451–462, 2013.
View at: Google Scholar
M. J. Borah and B. Hazarika, “Some results of mixed fuzzy soft topology and applications in chemistry,” Annals of Fuzzy Mathematics and Informatics, vol. 12, no. 1, pp. 129–138, 2016.
View at: Google Scholar
M. Riaz and M. R. Hashmi, “Fuzzy parameterized fuzzy soft topology with applications,” Annals of Fuzzy Mathematics and Informatics, vol. 13, no. 5, pp. 593–613, 2017.
View at: Publisher Site | Google Scholar
A. Khan, M. Izhar, and K. Hila, “On algebraic properties of DFS sets and its application in decision making problems,” Journal of Intelligent & Fuzzy Systems, vol. 36, no. 6, pp. 6265–6281, 2019.
View at: Publisher Site | Google Scholar
A. Khan, M. Izhar, and M. M. Khalaf, “Generalised multi-fuzzy bipolar soft sets and its application in decision making,” Journal of Intelligent & Fuzzy Systems, vol. 37, no. 2, pp. 2713–2725, 2019.
View at: Publisher Site | Google Scholar
M. Gulistan, N. Yaqoob, Z. Rashid, F. Smarandache, and H. A. Wahab, “A study on neutrosophic cubic graphs with real life applications in industries,” Symmetry, vol. 10, no. 6, p. 203, 2018.
View at: Publisher Site | Google Scholar
M. Gulistan, I. Beg, and N. Yaqoob, “A new approach in decision making problems under the environment of neutrosophic cubic soft matrices,” Journal of Intelligent & Fuzzy Systems, vol. 36, no. 1, pp. 295–307, 2019.
View at: Publisher Site | Google Scholar
K. Ullah, T. Mahmood, Z. Ali, and N. Jan, “On some distance measures of complex Pythagorean fuzzy sets and their applications in pattern recognition,” Complex & Intelligent Systems, vol. 6, no. 1, pp. 15–27, 2020.
View at: Publisher Site | Google Scholar
D. Alghazzawi, U. Shuaib, T. Fatima, A. Razaq, and M. A. Binyamin, “Algebraic characteristics of anti-intuitionistic fuzzy subgroups over a certain averaging operator,” IEEE Access, vol. 8, pp. 205014–205021, 2020.
View at: Publisher Site | Google Scholar
C. Bejines, M. J. Chasco, and J. Elorza, “Aggregation of fuzzy subgroups,” Fuzzy Sets and Systems, vol. 418, pp. 170–184, 2021.
View at: Publisher Site | Google Scholar
S. Dogra and M. Pal, “Picture fuzzy subgroup,” Kragujevac Journal of Mathematics, vol. 47, no. 6, pp. 911–933, 2023.
View at: Google Scholar
H. Alolaiyan, U. Shuaib, L. Latif, and A. Razaq, “T-intuitionistic fuzzification of Lagrange’s theorem of t-intuitionistic fuzzy subgroup,” IEEE Access, vol. 7, pp. 158419–158426, 2019.
View at: Publisher Site | Google Scholar
M. Akilesh and R. Muthuraj, “Properties of multi anti L-fuzzy quotient group A of a group G,” The Journal of Mathematics and Computer Science, vol. 12, 2022.
View at: Google Scholar
S. Bhunia, G. Ghorai, Q. Xin, and F. I. Torshavn, “On the characterization of Pythagorean fuzzy subgroups,” AIMS Mathematics, vol. 6, no. 1, pp. 962–978, 2021.
View at: Publisher Site | Google Scholar
H. Alolaiyan, H. A. Alshehri, M. H. Mateen, D. Pamucar, and M. Gulzar, “A novel algebraic structure of (α, β)-complex fuzzy subgroups,” Entropy, vol. 23, no. 8, p. 992, 2021.
View at: Publisher Site | Google Scholar
D. A. Kattan, M. Amin, and A. Bariq, “Certain structure of Lagrange’s theorem with the application of interval-valued intuitionistic fuzzy subgroups,” Journal of Function Spaces, vol. 2022, 9 pages, 2022.
View at: Publisher Site | Google Scholar
A. Imtiaz, U. Shuaib, H. Alolaiyan, A. Razaq, and M. Gulistan, “On structural properties of-complex fuzzy sets and their applications,” Complexity, vol. 2020, 13 pages, 2020.
View at: Publisher Site | Google Scholar
K. R. Balasubramanian and R. Revathy, “Product of (𝜆, 𝜇)-multifuzzy subgroups of a group,” Annals of the Romanian Society for Cell Biology, vol. 25, no. 6, pp. 2448–2460, 2021.
View at: Google Scholar
K. Ahmad, M. Bal, and M. Aswad, “The kernel of fuzzy and anti-fuzzy groups,” Journal of Neutrosophic and Fuzzy Systems, vol. 1, no. 1, pp. 48–54, 2022.
View at: Google Scholar

Copyright

Copyright © 2022 Ibtisam Masmali et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

424

Downloads

515

Citations